ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
L1
4
Int
e
gr
at
ion
3
The
fundam
e
nt
al
Theor
em
of
Calculus
Theor
em
Sup
p
ose
that
the
funct
ion
f
is
conti
nuous
on
an
int
er
v
al
I
cont
aining
th
e
poi
n
t
a.
PA
R
T
2
.
Defi
ne
the
funct
ion
fo
r
va
r
i
a
b
l
e
x
=
I
by
11
Fix)
t
Saf
itsd
t
,
x
=
I
.
Th
e
n
F
is
diff
e
r
e
nt
iable
on
I
,
and
Fix
=
fix
)
.
So
F
is
an
an
t
i
d
e
r
i
v
at
i
v
e
of
f on
1
:
12
)
*
)
,
"
fit
s
dt
=
fix
)
.
Pa
r
t
I
.
If
GN)
is
an
y
antideriv
ativ
e
of
fix
e
on
I
,
so
th
at
Gi
x
s
=
fix
)
on
1
,
then
fo
r
any
b
o
n
I
,
we
ha
v
e
(3)
Saf
(x
d
x
=
G1
b)
-G(
a)
.
Remark
:
0
Fix
)
is
a
funct
ion
on
X
.
โก
The
theo
r
em
is
true
be
caus
e
the
Mean
va
l
u
e
Th
e
or
e
m
of
int
egr
ation
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
โข
The
theo
r
em
is
the
mos
t
impor
t
ant
to
o
l
fo
r
the
cal
cul
at
i
on
of
int
egr
ation
.
โฃ
We
de
fine
the
ev
a
l
u
a
t
i
o
n
symb
o
l
Fixs
/
!
f
=(
b
)
-
F1a)
,
and
use
the
symb
o
l
/f
ixid
x
fo
r
the
indefinit
e
int
egr
al
,
i
.
e
.
gener
al
antideriv
ativ
e
of f
.
Then
B3)
be
com
e
s
S
!
fix
s
dx
=
((f
ix
dx
)
)
a
Pr
oof
.
We
use
the
de
finit
ion
of
de
riv
at
iv
e
F(xth
)
-
FIX)
F(
x
)
=
h
=
(S
**
fit
s
at
-
J
:
"
fit
s
at)
=
.
(
***
fit
s
at
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
-
lim
โ
.
(xth-
x)
ficl
with
c
+x
.
xth]
ยท
A
nt
o
Her
e
we
use
the
Mean
va
l
u
e
Th
eor
em
fo
r
int
egr
atio
n
.
As
h
to
,
we
hav
e
fix
l
->
fix
.
so
abo
v
e
limit
im
.
hf(x)
=
lim
f(
x
)
=
fix
,
c
->
X
4
wher
e
we
use
the
fa
c
t
that
fix
e
is
co
ntinuo
us
fo
r
the
last
equality
.
So
we
check
ed
that
F(x)
=
f(
x
)
,
which
is
(2)
.
To
check
PA
R
T
I
,
we
obs
e
r
v
e
that
G(
x
)
=
f(
x
)
,
so
F(
x
)
=
DIX)
+
C
fo
r
som
e
const
ant
C
.
So
fit
dt
=
FI
x)
=
GI
X)
+C
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Put
x
=
a
int
o
abo
v
e
,
0
=
(i
fit
)
d
+
=
F(
a)
=
G(
a)
+
C
=>
c
=
-
G1
a)
.
so
S
.
fit
dt
=
G(
x
)
+)
=
G(
x
)
-
G(
a)
.
Le
t
x=
b
,
we
no
w
pr
o
v
ed
(3)
(a
fit
)
dt
=
a(b)
-
G(
a)
=
G(
x
)
/
a
-
i
-
Ex
ample
.
Find
the
ar
e
a
fo
r
the
re
g
i
o
n
=
41x
.
Y
GIR
Y
)
8
*
Y
*
Sin
x
3
.
so
lu
tio
n
.
Ar
ea
=I
*
sin
x
d
x
=7
c0sx>
ยท
xY
=
f(
0
)
x
)
-
-1
0
s
O
)
A
=
Sinx
โ
-
=
-
(
-
1)
+
1
=
2
.
B
<X
X
I
Fc
o
s
X
)
'
=
sin
x
1
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Ex
ample
.
Find
the
de
riv
at
iv
e
s
of
the
fo
l
l
o
w
i
n
g
funct
ions
:
cal
Fix
=
/
2
-
*
dt
,
(b)
G(
x)
=
x
5x
e-
t
d
t
,
ยท
14
14
Hix
=
I
e-
t
at
.
so
lu
tio
n
.
We
ur
e
the
fundam
e
nt
al
theo
r
em
.
(a)
F(
x
)
=
- S
,
e
-t
dt
So
Fix
=
-
(S
,
e
+
dt
)
=
-
e
-
x
-
-
T
fundam
e
nt
al
theor
em
fo
r
fix
)
=
e
*
(a)
<Metho
d
2)
FIx)
=
-S
,
e
+i
t
=/
,
)
-
et]
dt
we
use
fundam
ent
al
the
or
e
m
fo
r
fix
=
-e
*
th
e
n
Fix)
=
-
e
-
*
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
(b
)
The
calculation
is
giv
en
by
pr
o
d
uctio
n
rule
fo
r
de
riv
at
iv
e
.
(ax))
=
x2
)
.
et't
+
x2)
ta
t
)
'
ยท
5
=
2x
.
j
et'at
+
x2
e-
15x
-
cha
in
rule
x
e-
t
d
t
-
1
*
et'
d
t
(2)
HI
x
=
So
(Hixs)'
=
exe
*
==
x
e
ยท
2X
-
-
x
*
=
3x
x
-
2x
-
e
*
(
*
*
fit
s
dt
=
fit
)
+
=
qx
.
9ix)
=
f(
9
,
x))
.
gix
)
.
*
-
fit
d+
=
f(
gx
)
gix
-
f(
hix
)
hix
)
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Int
e
gr
at
ion
Ta
b
l
e
Po
l
y
n
o
m
a
i
l
type
:
(x
ad
x
=
+x
**
+
c(a+
-
1)
/x
+
dx
=
Jedx
=
en(x)
+
c
Tr
i
g
e
o
m
e
t
r
i
c
type
:
/
sin(ax)
dx
=
-
I
cos
(
ax
)
+
> (a
=
0)
S
coslaxsdx
=
si
n
(
ax
s
+
> (a
F
0)
I
secsaxid
x
=
โ
ta
n
l
a
x
s
+
>
(at
o
)
Hyper
trigeo
metuc
type
:
/sin
h(axsd
x
=
cosh(axs
+
<
(a
F0
)
Sco
sh(ax)
&x
=
sin
h
(ax)
+
>
(a
Fo
)
Inv
e
r
s
e
of
trigeo
metric
type
:
S
Idx
=
sin
"
( )
+
>
(a
<o
)
a
-
x2
I
ar
t
ar
dx
=
I
ta
n
"
)
+
C
.
Expo
nential
type
sea
"
d
x
=
e4+C
.
(b
**
dx
=
a
!
s
b
**
+
C
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
chain
rule
and
method
of
su
b
stitu
tio
n
Recall
the
chain
rule
Po
g
)
(
x
)
=
&(f(9
/
x)))
=
fig
,
x)
-
gix
)
.
So
we
int
egr
at
e
abo
v
e
equation
an
d
get
(f
(gix)
ยท
gix
dx
=
fig,x
)
+
C
We
let
u=
gixs
.
Then
du
d
X
=
giX)
.
so
in
diff
e
r
e
nt
ial
fo
r
m
du=
gix
dx
.
If'(
9
(
x
)
)
-
gix
dx
=
/f
i(u)d
u
=
f(
u)
+
C
=
f(
9
1
x)
)
+
C
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Ex
ample
I
sin
(3
enx)
dx
X
so
lutio
n
.
Le
t
U
=
ex
.
Then
du=
-
dx
/Sin
(3
enx)
dx
=
/
Sin
(3
us
du
=S
sin
(3
us
d (
34
)
=
(d
)-
co
ssn)
=-
ws(u
l
+
C
=
cos(1x)
+
C
i
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
โค
I
n
t
eg
r
atio
n
by
par
ts
su
ppo
se
that
U(X
)
and
VIx>
ar
e
tw
o
diff
e
r
e
nt
iable
funct
ions
.
The
pr
o
d
u
ct
Rule
=>
*
(UI
x)
.
X(
x
)
)
=
Mix a
+
xix)
Y
By
int
egr
ating
bot
h
sid
e
and
mo
ving
te
r
m
s
,
we
obt
ain
Jucx>
x
=
ux
)
.
vix)
-
/vix)
d
*
We
also
writ
e
abo
v
e
by
SUNV=U
.
V-
S
v
d
u
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Ex
ample
I
x
ed
x
so
lu
tio
n
Le
t
U
=
X
dy
=
exd
x
=
de
x
(x
e
x
d
x
=
/
x d
(e
Y)
=
Judy
=UV
-
Sv
d
U
=
x
.
ex
-
Sed
x
=
x
.
e
-
e
+
C
I
Ex
ample
.
Seux
d
x
=
JUdy
U=eux
.
V
=
x
=
U
.
X-
S
V
d
U
=
(n
x)
.
X-
X
-
I
dx
=x
1
x
-
x
+
C(x
>
0)
1
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Summar
y
of
To
d
a
y
:
*
)
fit
dt
f(
x
)
mean
va
l
u
e
Theor
em
Ja
f
(
t
)
a
t
=
(d
)
/
comput
ation
To
o
l
in
de
finit
e
int
egr
al
or
antideriv
ativ
e
chain
rule
("
co
mpo
sitio
n
rule")
(f
(gix)
ยท
gix
dx
=
fig,x
)
+
C
Int
e
gr
at
ion
by
par
t
("pr
o
d
uctio
n
rule"
(
Sud
y
=
n
.
V-
S
u
d
u
Int
e
gr
at
ion
ta
b
l
e
.